Snapping vertex to face during proportional edit
Besides all the good suggestions that came so far, what comes closest to your question in my opinion is the project individual option of Blender. You need to set snapping to snap to face first to see the button. Then you can toggle this option on in the 3D Viewport (red square):



Now, all translation tools, move, rotate, scale, even with proportional editing, will snap the whole selection to other faces from the view you're looking at (hold down Ctrl to enable snapping of course):



Quick animation / comparison here: